Matrix algebra: questions for repetition

Matrix algebra: questions for repetition

Matrix algebra is a bunch of dreadful rules is all that many students remember after studying it. It's a relief to know that a simple property like is more important than remembering how to calculate an inverse. The theoretical formula for the inverse at the level of this course is not necessary; if you need to find the inverse of a numerical matrix, you can use Excel.

First things first

Three big No's: 1) there is no commutativity, in general, 2) determinants don't apply to non-square matrices, and 3) don't try to invert a non-square matrix. There are ways around these problems but all of them are deficient, so better stick to good cases.

Three big ideas: 1) the analogy with real numbers is the best guide to study matrices, 2) the matrix product definition is motivated by the desire to compactify a system of equations, 3) symmetric matrices have properties closest to those of real numbers.

Three surprises: 1) in general, matrices don't commute (can you give an example?), 2) a nonzero matrix is not necessarily invertible (can you give an example?), 3) when you invert a product, you have to change the order of the factors (same goes for transposition) These two properties are called reverse order laws.

Properties of the identity matrix: 1) use the definition of the inverse to find the inverse of the identity matrix, 2) do you think the identity matrix commutes with any other matrix? 3) Can you name any matrices, other than the identity, satisfying the equation If a matrix satisfies this equation, what can you say about its determinant? 4) What is the determinant of the identity matrix?

If a nonzero number is close to zero, then its inverse must be a large number (in absolute value). True or wrong? Can you indicate any analogs of this statement for matrices?

Suppose matrices are given and How would you solve the linear matrix equation for

Symmetric matrices: 1) For any matrix both matrices and are symmetric. True or wrong? 2) If a matrix is symmetric and its inverse exists, will the inverse be symmetric?